HL congruence
/ aitch-el /
HL is a congruence criterion reserved for right triangles. It says: if the hypotenuse and one leg of one right triangle are equal to the hypotenuse and one leg of another right triangle, the two are congruent. (The right angle is the shared starting condition; HL is the extra information.) In some countries this rule is taught as RHS — right angle, hypotenuse, side — naming the same thing.
Precisely: suppose triangles ABC and DEF each have a right angle (say at C and at F), with hypotenuses AB and DE. If |AB| = |DE| (hypotenuses) and one pair of legs is equal, say |BC| = |EF|, then triangle ABC is congruent to triangle DEF. Why is this allowed when general 'SSA' is not? Because the Pythagorean theorem fills in the third side: the missing leg is the square root of (hypotenuse squared minus known leg squared), so both triangles are forced to have identical third sides, and SSS then closes the case. The right angle is exactly what removes the SSA ambiguity.
Treat HL as the one safe member of the SSA-looking family: it works only because of the right angle. Without that 90-degree guarantee, two sides and a non-included angle do not determine a triangle, so never extend HL to non-right triangles.
Two right triangles each have a hypotenuse of 13 and one leg of 5. By HL they are congruent; indeed the other leg in each is the square root of (13^2 - 5^2) = square root of 144 = 12, so all three sides match.
Right angle plus equal hypotenuse and one leg forces congruence.
HL applies only to right triangles. It is the lone safe case of the otherwise invalid SSA pattern, rescued by the right angle and the Pythagorean theorem.